This invention provides means for improving control of continuous processes that handle liquids, and therefore provides benefits to manufacturers by enabling them to effectively monitor and operate their processes. Data generated by this invention can be used to control the gas contents of liquids within optimum ranges, for instance in paper coating processes and in the manufacture of such products as food products (ketchup, mayonnaise, syrup), personal care products (skin cream, shampoo), pharmaceutical products, paints, petroleum blends, and the like. This invention is useful in any industry where information on entrained and/or dissolved air and other gases, and related parameters such as true density of and gas solubility in process liquids, is employed to optimize processing.
Those skilled in the arts of processing liquids desire to know how much air and/or other gases are entrapped and dissolved therein for a variety of reasons. Entrapped air can cause undesired foaming during processing, e.g. in papermaking and in the preparation of foodstuffs, and can result in disruption of film products, e.g. from paints. Entrained gases distort such processing parameters as density, making precise control of processes impossible. Those skilled in the art know that, generally, the more viscous a fluid being processed, the more difficult it is for any entrained air to escape from it and consequently the greater the amount of air bubbles likely to be accumulated therein. Also, as pressure on a fluid is lowered, dissolved air or other gas therein tends to leave solution and form bubbles in the fluid.
There are a number of instruments that are currently commercially available for measuring the air or gas content in a liquid. Such instruments include Valmet""s COLORMAT, Mxc3xctek""s GAS-60, Papec""s PULSE))))AIR, Capella Technology""s CAPTAIR, Anton-Paar""s CARBO 2100 CO2 analyzer, and CyberMetrics"" AIR TESTER.
Mxc3xctek""s GAS-60, for instance, is said to be useful in the context of minimizing pinholes (voids) in papermaking processes. Pinholes develop when pressure is reduced and dissolved gasesxe2x80x94which accumulate in the papermaking process due to mechanical effects and chemical and biological reactionsxe2x80x94are released. The GAS 60 is installed on line and is used to determine the gas content of entrained and dissolved gases in pulp suspensions. Having determined gas content, process engineers are able to calculate how much (expensive) deaerating additive should be used, and thus to avoid unnecessarily increased manufacturing costs due to employing too much deaerating additive.
Papec""s PULSE))))AIR_V3 is a sensor for the measurement of entrained air and gases in process fluids. It is said to be useful in the pulp and paper industry in connection with machine headboxes and white water systems, coatings, and brownstock washers, in the secondary fiber industry (for effluent treatment), in the paint industry, in oil bottling processes, in the processing of well drilling muds, and in general in any application needing entrained air information.
Anton-Paar""s CARBO 2100 CO2 analyzer employs a patented impeller method which is said to make it significantly faster that other commercially available systems for measuring and monitoring tasks and also for regulating the CO2 content of process liquids during production runs in the beer and soft drink industry.
It is believed that all of these instruments adopt a common approach, using Boyle""s Law. Boyle""s law is given by the formula
P1V1=P2V2xe2x80x83xe2x80x83(1) 
where V1 and V2 are the volumes of the free gas in the liquid at two different pressures, P1 and P2, respectively. Being a xe2x80x9ctwo-point measurementxe2x80x9d, this common approach measures the volume difference xcex94V=V1xe2x88x92V2 between P1 and P2, and calculates the volumes of free gas, V1 and V2, from Boyle""s Law as                               V          1                =                                                                              P                  2                                ⁢                Δ                ⁢                                  xe2x80x83                                ⁢                V                                                              P                  2                                -                                  P                  1                                                      ⁢                          xe2x80x83                        ⁢            and            ⁢                          xe2x80x83                        ⁢                          V              2                                =                                                    P                1                            ⁢              Δ              ⁢                              xe2x80x83                            ⁢              V                                                      P                2                            -                              P                1                                                                        (        2        )            
More general formulas, which correlate the volumes of free gas with the pressures being acted upon, can be derived from the Ideal Gas Law as
P1V1=n1RT1xe2x80x83xe2x80x83(3) 
and
P2V2=n2RT2xe2x80x83xe2x80x83(4) 
where R is the gas constant, and n1, T1 and n2, T2 are moles of free gas and temperatures at P1 and P2, respectively. In the case of n1=n2 and T1=T2, equations (3) and (4) can be simplified to the equation of Boyle""s Law given in (1). Hence, Boyle""s Law is, in fact, a special case of the Ideal Gas Law and is valid only if the moles of free gas and temperatures at P1 and P2 are kept constant.
In practice, a portion of free gas, however, will be dissolved into the liquid. The solubility of gas is, as a general rule, proportional to the gas pressure as stated in Henry""s Law
P=Hndxe2x80x83xe2x80x83(5) 
where P, H, nd are the pressure of the gas being dissolved, the constant of Henry""s Law, and moles of dissolved gas, respectively. This unquestionably makes n1xe2x89xa0n2 between P1 and P2, causing a violation of Boyle""s Law. Therefore, using Boyle""s Law for a xe2x80x9ctwo-point measurementxe2x80x9d is an unreliable approximation and can cause a significant amount of error, especially when the pressure difference between the two points becomes large.
To cure this error, there have been some attempts to use Henry""s Law to compensate for the amount of the dissolved gas. This approach, however, is generally impractical, inasmuch as the constants of Henry""s Law are not available for many process liquids, particularly for those containing multiple-components such as coating slurries. Using the known constant of one liquid to approximate the constant of the others may potentially introduce a considerable amount of error, because the solubility of gases such as air changes dramatically from liquid to liquid. The solubility of air in isooctane at standard temperature and pressure, for example, is more than 100 times higher than the solubility of air in water.
The present invention provides methods and apparatuses for determining the entrained gas content and/or the dissolved gas content of liquids. This invention provides means for improving control of continuous processes that handle liquids, and therefore provides benefits to manufacturers by enabling them to effectively monitor and operate their processes. Data generated by this invention can be used for instance to control the gas contents of liquids within optimum ranges, for instance in the processing of foam such as shaving cream or ice cream and to minimize gas contents, for instance in paper coating processes and in the manufacture of such products as food products (ketchup, mayonnaise, syrups, various sauces), personal care products (skin cream, shampoo, lotions, toothpaste), pharmaceutical products, herbicides, paints, lubricating greases, petroleum blends, water softeners, and the like. This invention is useful in any industry where information on entrained and/or dissolved gas, and related parameters such as true density and solubility of process liquids, is employed.
In one embodiment, this invention provides a method for controlling the entrained gas content of a liquid or slurry being flow-processed. The liquid or slurry being flow-processed may bexe2x80x94without limitationxe2x80x94a slurry of kaolin clay, calcium carbonate, titanium dioxide, or alumina trihydrate being supplied as a coating to a paper substrate. Alternativelyxe2x80x94again without limitationxe2x80x94the liquid or slurry being flow-processed is ointment, cream, lotion, toothpaste, mayonnaise, ketchup, or lubricating grease being packaged into a retail container. The method comprises: a.) setting a quantitative target for the free gas content of said liquid or slurry; b.) continuously flowing said liquid or slurry and mixing an antifoam agent therewith; c.) determining the volume percentage of free gas, x %, from the formula       x    ⁢          xe2x80x83        ⁢    %    =            V      s                      V        s            +      V      
wherein Vs is the volume of free gas under standard conditions and V is the gas-free volume of the liquid carrier component; e.) comparing the calculated free gas content to the target free gas content; and, f.) if the calculated free gas content is greater than the target free gas content, raising the amount of antifoam agent mixed in step b.).
In another embodiment, this invention provides an apparatus comprising a reservoir for process fluid, piping through which the process fluid may be pumped, and a deaerator unit, the improvement which comprises locating means for detecting the volume percentage of free gas in the process fluid in working relationship to the deaerator unit. In this apparatus, a single such detection means may be located downstream of the deaerator unit and wherein the apparatus further comprises an alarm capable of signaling the presence in the process fluid of a volume percentage of free gas higher than a pre-specified level. Alternatively, the apparatus may include two such detection means located, respectively, upstream and downstream of the deaerator unit and the apparatus may further comprises a comparator capable of determining and indicating the magnitude of any difference in volume percentage of free gas in the process fluid upstream of and downstream of the deaerator unit.
Another embodiment of this invention is a method for determining the amount of gas in a liquid which comprises subjecting a mixture of an incompressible liquid sample and a compressible gas to at least three different equilibrium pressure states, measuring the temperature and volume of the mixture at each of the at least three pressure states, determining the changes in volume of the mixture between at least two different pairs of pressure states, and calculating the amount of gas in the liquid sample by using the equation       x    ⁢          xe2x80x83        ⁢    %    =            V      s                      V        s            +      V      
wherein V is the volume of the gas-free liquid in a sample chamber at ambient pressure and Vs is determined by the equation                               V          s                =                ⁢                                                            Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  1                                ⁢                                  T                  s                                            T                        ⁢                                          P                1                2                                                              P                  s                                ⁡                                  (                                                            P                      2                                        -                                          P                      1                                                        )                                                              +                                                    Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  2                                ⁢                                  T                  s                                            T                        ⁢                                                            P                  3                                ⁡                                  (                                                            P                      2                                        -                                          P                      1                                                        )                                                                              P                  s                                ⁡                                  (                                                            P                      3                                        -                                          P                      2                                                        )                                                              +                                                ⁢                                                            P                1                            -                              P                s                                                    P              s                                [                                                    Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  1                                ⁢                                  T                  s                                            T                        -                                                            Δ                  ⁢                                      xe2x80x83                                    ⁢                                      V                    2                                    ⁢                                      T                    s                                                  T                            ⁢                                                                    P                    3                                    ⁡                                      (                                                                  P                        2                                            -                                              P                        1                                                              )                                                                                        P                    1                                    ⁡                                      (                                                                  P                        3                                            -                                              P                        2                                                              )                                                                                ]                    
wherein P1, P2, and P3 are three different equilibrium ambient pressures, Ps and Ts are standard pressure and temperature, xcex94V1 and xcex94V2 are the volume difference of the free air measured at an equilibrium state between P1 and P2 and P2 and P3, respectively. In preferred embodiments of this embodiment of the invention, the at least three equilibrium pressure states differ from one another at least to the extent that the three different volumes differ from one another by at least 0.1% and/or the at least three equilibrium pressure states differ from one another at least to the extent that three different apparent densities of said liquid differ from one another by at least 0.1%. In other preferred embodiments of this embodiment of the invention, the at least three pressure states differ from one another by at least 0.1 psi and/or the at least three pressure states differ from one another by at least 1 atmosphere. In this embodiment of the invention, the determination of changes in volume may be accomplished by measurement of volumes or by measurement of apparent densities.
The present invention also provides an apparatus that includes: a reservoir for process fluid; piping through which the fluid may be pumped, said piping being under the control of a pressure regulator which is capable of setting at least three different pressures P1, P2, and P3 in the apparatus; at least three fluid control valves V1, V2, and V3; a pressure gauge; a temperature gauge; and a density gauge. This apparatus may be used for obtaining data for use in determining amounts of air entrained or dissolved in a fluid, by a method that comprises: providing the apparatus as described; collecting stabilized pressure, temperature, and density data at a first pressure level; collecting stabilized pressure, temperature, and density data at a second pressure level; and collecting stabilized pressure, temperature, and density data at a third pressure level.
Yet another embodiment of this invention is a method for automatically controlling the output of a continuous process that requires mixing of a solid or liquid component with a liquid carrier component. This method embodiment of the invention comprises: a.) setting a quantitative target for weight-% of one or more solids and/or concentration of one or more liquids to the liquid carrier component; b.) continuously mixing said solids and/or liquids with the liquid carrier component; c.) filling a measurement chamber with the blended mixture and allowing it to come to equilibrium; d.) recording equilibrium temperature, T1, the volume of the sample, V1, and equilibrium pressure, P1; e.) increasing or decreasing the volume of the mixture in the sample chamber, allowing the fluid to come to equilibrium, and recording equilibrium temperature, T2, sample volume, V2, and equilibrium pressure, P2; f.) again, increasing or decreasing the volume of the mixture in the sample chamber, allowing the fluid to come to equilibrium, and recording equilibrium temperature, T3, sample volume, V3, and equilibrium pressure, P3; g.) determining the true density, xcfx81, by employing the formula   ρ  =      m    V  
wherein the mass, m, is the mass of the liquid mixture sample, and, the gas-free volume of the liquid mixture, V, and the volume percentage of free air or other gas, x %, are calculated from the formulas   V  =            V      t1        -          [                        Δ          ⁢                      xe2x80x83                    ⁢                      V            1                    ⁢                                    P              1                                      (                                                P                  2                                -                                  P                  1                                            )                                      +                  Δ          ⁢                      xe2x80x83                    ⁢                      V            2                    ⁢                                                    P                3                            ⁡                              (                                                      P                    2                                    -                                      P                    1                                                  )                                                                    P                1                            ⁡                              (                                                      P                    3                                    -                                      P                    2                                                  )                                                        ]      
and,       x    ⁢          xe2x80x83        ⁢    %    =            V      s                      V        s            +      V      
wherein xcex94V1 is the volume difference of the free gas between P1 and P2, xcex94V2 is the volume difference of the free gas between P2 and P3, Vt1 is the total volume of the liquid and entrained gas, and Vs is the volume of free air or other gas under standard conditions and is calculated from the formula                               V          s                =                ⁢                                                            Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  1                                ⁢                                  T                  s                                            T                        ⁢                                          P                1                2                                                              P                  s                                ⁡                                  (                                                            P                      2                                        -                                          P                      1                                                        )                                                              +                                                    Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  2                                ⁢                                  T                  s                                            T                        ⁢                                                            P                  3                                ⁡                                  (                                                            P                      2                                        -                                          P                      1                                                        )                                                                              P                  s                                ⁡                                  (                                                            P                      3                                        -                                          P                      2                                                        )                                                              +                                                ⁢                                                            P                1                            -                              P                s                                                    P              s                                [                                                    Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  1                                ⁢                                  T                  s                                            T                        -                                                            Δ                  ⁢                                      xe2x80x83                                    ⁢                                      V                    2                                    ⁢                                      T                    s                                                  T                            ⁢                                                                    P                    3                                    ⁡                                      (                                                                  P                        2                                            -                                              P                        1                                                              )                                                                                        P                    1                                    ⁡                                      (                                                                  P                        3                                            -                                              P                        2                                                              )                                                                                ]                    
wherein these variables are provided by the data collected in steps d.-f.), and standard conditions refer to P=Ps=1 atm, T=Ts=273 K; h.) calculating the weight-% of solids and/or the liquid concentration in the mixture with the equation   m  =            1              (                  1          -                                    ρ              L                        ⁢                                          ∑                                  i                  =                  1                                n                            ⁢                                                                    k                    i                                    ⁢                                      x                    i                                                                                        (                                          ρ                      s                                        )                                    i                                                                    )              -                            ρ          L                          (                      1            -                                          ρ                L                            ⁢                                                ∑                                      i                    =                    1                                    n                                ⁢                                                                            k                      i                                        ⁢                                          x                      i                                                                                                  (                                              ρ                        s                                            )                                        i                                                                                )                    ⁢              (                  1          ρ                )            
wherein xcfx81L is the density of the liquid carrier component, ki is the Additive Volume Coefficient (AVC) for each solid or liquid, xi is the weight-% dry for each solid or the concentration for each liquid, (xcfx81s)i is the density of each solid or liquid, and xcfx81 is the true density of the mixture; i.) comparing the calculated weight-% solids or concentration to the target weight-% solids or concentration; and, j.) if the calculated weight-% solids or concentration is greater or less than the target weight-% solids or concentration, lowering or raising the amount of solids or liquids mixed in step b.).
Another inventive method for automatically controlling the output of a continuous process that requires mixing of a solid or liquid component with a liquid carrier component of this invention includes the steps of: a.) setting a quantitative target for weight-% of one or more solids and/or concentration of one or more liquids to the liquid carrier component; b.) continuously mixing said solids and/or liquids with the liquid carrier component; c.) diverting a fluid sample from the main piping system into the sample measurement chamber and allowing the sample to come to equilibrium; d.) recording equilibrium temperature, T1, equilibrium density, xcfx811, and equilibrium pressure, P1; e.) adjusting the pressure of the fluid in the sample chamber, allowing the fluid to come to equilibrium, and recording equilibrium temperature, T2, equilibrium density, xcfx812, and equilibrium pressure, P2; f.) again, adjusting the pressure of the fluid in the sample chamber, allowing the fluid to come to equilibrium, and recording equilibrium temperature, T3, equilibrium density, xcfx813, and equilibrium pressure, P3; g.) determining the true density, xcfx81, by employing the formula   ρ  =      1    V  
wherein the volume, V, is calculated from the formula   V  =            1              ρ        1              -          [                                    (                                          1                                  ρ                  1                                            -                              1                                  ρ                  2                                                      )                    ⁢                                    P              1                                      (                                                P                  2                                -                                  P                  1                                            )                                      +                              (                                          1                                  ρ                  2                                            -                              1                                  ρ                  3                                                      )                    ⁢                                                    P                3                            ⁢                              (                                                      P                    2                                    -                                      P                    1                                                  )                                                                    P                1                            ⁢                              (                                                      P                    3                                    -                                      P                    2                                                  )                                                        ]      
wherein these variables are provided by the data collected in steps d.-f.); h.) calculating the weight-% of solids and/or the liquid concentration in the mixture with the equation   m  =            1              (                  1          -                                    ρ              L                        ⁢                                          ∑                                  i                  =                  1                                n                            ⁢                                                                    k                    i                                    ⁢                                      x                    i                                                                                        (                                          ρ                      S                                        )                                    i                                                                    )              -                            ρ          L                          (                      1            -                                          ρ                L                            ⁢                                                ∑                                      i                    =                    1                                    n                                ⁢                                                                            k                      i                                        ⁢                                          x                      i                                                                                                  (                                              ρ                        S                                            )                                        i                                                                                )                    ⁢              (                  1          ρ                )            
wherein xcfx81L is the density of the liquid carrier component, ki is the Additive Volume Coefficient (AVC) for each solid or liquid, xi is the weight-% dry for each solid or the concentration for each liquid, (xcfx81s)i is the density of each solid or liquid, and xcfx81 is the true density of the mixture; i.) comparing the calculated weight-% solids or concentration to the target weight-% solids or concentration; and, j.) if the calculated weight-% solids or concentration is greater or less than the target weight-% solids or concentration, lowering or raising the amount of solids or liquids mixed in step b.).
Both of these two methods of this invention can be used in a process for continuously coating a substrate, for instance where the substrate is a paper web and the solids component comprises kaolin clay, calcium carbonate, titanium dioxide, or alumina trihydrate, in a method that comprises: a.) setting a quantitative target for weight-% of one or more solids to be coated onto a substrate; b.) continuously applying the solids to the substrate via a carrier fluid; c.) measuring the apparent density of the slurry; d.) determining the true density of the slurry; e.) calculating the weight-% of solids in the slurry in the manner either of the former or the latter general method; f.) comparing the calculated weight-% solids to the target weight-% solids; and, g.) if the calculated weight-% is greater or less than the target weight-%, lowering or raising the amount of solids applied in step b.).
Likewise, either of these two general methods of this invention can be used in a process for controlling the output of a continuous process for preparing a syrup. This method comprises: a.) setting a quantitative target for a concentration of one or more carbohydrates, e.g. sucrose, and/or carbohydrate-containing liquids, e.g. corn syrup and high fructose corn syrup, to be blended, along with water, into a syrup; b.) continuously supplying the carbohydrate and/or carbohydrate-containing liquid and a dilution liquid to a vessel and mixing said liquids to form a slurry; c.) measuring the apparent density of the slurry; d.) determining the true density of the slurry; e.) converting this true density to the calculated carbohydrate concentration; f.) comparing the calculated carbohydrate concentration to the target carbohydrate concentration; and, g.) if the calculated carbohydrate concentration is greater or less than the target carbohydrate concentration, lowering or raising the amount of carbohydrates and/or volume of carbohydrate-containing liquids supplied in step b.).
In a third group of practical applications, this invention provides a method for controlling the output of a continuous process for preparing a carbonated beverage, e.g. a soft drink, beer, or a carbonated wine. This method comprises: a.) setting a quantitative target for a concentration of carbon dioxide to be blended into an aqueous medium; b.) continuously supplying carbon dioxide to the aqueous medium in a vessel and mixing those components to form a carbonated aqueous medium in the vessel at a preset xe2x80x9cbottlingxe2x80x9d pressure P0, wherein P0 is the produced xe2x80x9cbottlingxe2x80x9d pressure inside a sealed carbonated beverage container, at which pressure all of the free carbon dioxide is dissolved into the aqueous medium; c.) diverting a carbonated aqueous medium sample from the vessel into a sample measurement chamber at the same xe2x80x9cbottlingxe2x80x9d pressure P0; d.) reducing the aqueous medium pressure from P0 to P1 allowing the dissolved carbon dioxide to start to be released back to the aqueous medium in a free-bubble form; e.] reducing the aqueous medium pressure further from P1 to P2 allowing a sufficient amount of the dissolved carbon dioxide to be released back to the aqueous medium in a free-bubble form; f.) measuring the change in volume of the carbon dioxide liquid mixture at an equilibrium state between P1 and P2; g.) reducing the aqueous medium pressure further from P2 to P3 allowing more dissolved carbon dioxide to be released back to the aqueous medium in a free-bubble form; h.) measuring the change in volume of the carbon dioxide liquid mixture at an equilibrium state between P2 and P3; i.) determining the volume of free carbon dioxide, Vs, in the carbonated aqueous medium at the standard condition using the equation       V    s    =                              Δ          ⁢                      xe2x80x83                    ⁢                      V            1                    ⁢                      T            s                          T            ⁢                        P          1          2                                      P            s                    ⁢                      (                                          P                2                            -                              P                1                                      )                                +                            Δ          ⁢                      xe2x80x83                    ⁢                      V            2                    ⁢                      T            s                          T            ⁢                                    P            3                    ⁢                      (                                          P                2                            -                              P                1                                      )                                                P            s                    ⁢                      (                                          P                3                            -                              P                2                                      )                                +                                        P            1                    -                      P            s                                    P          s                    ⁡              [                                            Δ              ⁢                              xe2x80x83                            ⁢                              V                1                            ⁢                              T                s                                      T                    -                                                    Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  2                                ⁢                                  T                  s                                            T                        ⁢                                                            P                  3                                ⁢                                  (                                                            P                      2                                        -                                          P                      1                                                        )                                                                              P                  1                                ⁢                                  (                                                            P                      3                                        -                                          P                      2                                                        )                                                                    ]            
wherein P1, P2, and P3 are three different equilibrium ambient pressures, Ps and Ts are standard pressure and temperature, xcex94V1, and xcex94V2 are the volume difference of the free carbon dioxide measured at an equilibrium state between P1 and P2 and P2 and P3, respectively; j.] calculating the carbon dioxide concentration using the equation       x    ⁢          xe2x80x83        ⁢    %    =            V      s                      V        s            +      V      
wherein Vs is the volume of free carbon dioxide determined in i.] and V is the volume of carbonated aqueous medium in the sample chamber at a preset xe2x80x9cbottlingxe2x80x9d pressure P0 upon which no free bubble should present; k.) comparing the calculated carbon dioxide concentration to the target carbon dioxide concentration; and, l.) if the calculated carbon dioxide concentration is greater or less than the target carbon dioxide concentration, lowering or raising the volume of carbon dioxide supplied in step b.).
This invention also provides various analytical methods. One is a method for determining the concentration of a solid or liquid component in a liquid carrier component, which comprises measuring the apparent density of the mixture; determining therefrom the true density of the mixture; and calculating the weight-% of solids in the slurry in the manner taught above. Another is a method for determining the true density of a solid or liquid component in a liquid carrier component, which comprises measuring the apparent density of the mixture; and determining therefrom the true density of the mixture. Another is a method for determining the entrained air content of a liquid component, which comprises measuring the apparent air content of the liquid at a variety of pressures; and determining therefrom the true entrained air content of the liquid. Another is a method of characterizing a liquid by determining its entrained air content, which comprises calculating the volume percentage of free air, x %, in the liquid using the equation       x    ⁢          xe2x80x83        ⁢    %    =            V      s                      V        s            +      V      
wherein V is the volume of the gas-free liquid in a sample chamber at ambient pressure and Vs is determined by the equation       V    s    =                              Δ          ⁢                      xe2x80x83                    ⁢                      V            1                    ⁢                      T            s                          T            ⁢                        P          1          2                                      P            s                    ⁢                      (                                          P                2                            -                              P                1                                      )                                +                            Δ          ⁢                      xe2x80x83                    ⁢                      V            2                    ⁢                      T            s                          T            ⁢                                    P            3                    ⁢                      (                                          P                2                            -                              P                1                                      )                                                P            s                    ⁢                      (                                          P                3                            -                              P                2                                      )                                +                                        P            1                    -                      P            s                                    P          s                    ⁡              [                                            Δ              ⁢                              xe2x80x83                            ⁢                              V                1                            ⁢                              T                s                                      T                    -                                                    Δ                ⁢                                  xe2x80x83                                ⁢                                  V                  2                                ⁢                                  T                  s                                            T                        ⁢                                                            P                  3                                ⁢                                  (                                                            P                      2                                        -                                          P                      1                                                        )                                                                              P                  1                                ⁢                                  (                                                            P                      3                                        -                                          P                      2                                                        )                                                                    ]            
wherein P1, P2, and P3 are three different equilibrium ambient pressures, Ps and Ts are standard pressure and temperature, xcex94V1 and xcex94V2 are the volume difference of the free air measured at an equilibrium state between P1 and P2 and P2 and P3, respectively.
Finally, this invention provides various methods of identifying samples of unknown constitution. One such method comprises comparing its entrained air content with a collection of entrained air contents, for a variety of known compounds, determined in the manner taught above. Another comprises comparing its true density with a collection of true densities, for a variety of known compounds, determined by use of the relationship   ρ  =            m      V        .  
Another comprises comparing its % solids with a collection of % solids, for a variety of known compounds, determined according to the relationship   m  =            1              (                  1          -                                    ρ              L                        ⁢                                          ∑                                  i                  =                  1                                n                            ⁢                                                                    k                    i                                    ⁢                                      x                    i                                                                                        (                                          ρ                      S                                        )                                    i                                                                    )              -                            ρ          L                          (                      1            -                                          ρ                L                            ⁢                                                ∑                                      i                    =                    1                                    n                                ⁢                                                                            k                      i                                        ⁢                                          x                      i                                                                                                  (                                              ρ                        S                                            )                                        i                                                                                )                    ⁢                        (                      1            ρ                    )                .            